不定积分∫sin2xdx过程 ∫sin2xdx=
\u4e0d\u5b9a\u79ef\u5206\u7684\u63a8\u5bfc\u8fc7\u7a0b\u222b \u221a(x²+1) dx
\u4ee4x=tanu\uff0c\u5219\u221a(x²+1)=secu\uff0cdx=sec²udu
=\u222b sec³u du
\u4e0b\u9762\u8ba1\u7b97
\u222bsec³udu
=\u222b secudtanu
=secutanu - \u222b tan²usecudu
=secutanu - \u222b (sec²u-1)secudu
=secutanu - \u222b sec³udu + \u222b secudu
=secutanu - \u222b sec³udu + ln|secu+tanu|
\u5c06- \u222b sec³udu\u79fb\u652f\u7b49\u5f0f\u5de6\u8fb9\u4e0e\u5de6\u8fb9\u5408\u5e76\u540e\u9664\u4ee5\u7cfb\u6570\u5f97\uff1a
\u222bsec³udu=(1/2)secutanu + (1/2)ln|secu+tanu| + C
\u222bsin2xdx=-1/2*cos2x+C\u3002\uff08C\u4e3a\u4efb\u610f\u5e38\u6570\uff09\u3002
\u89e3\u7b54\u8fc7\u7a0b\u5982\u4e0b\uff1a
\u222bsin2xdx
=1/2\u222bsin2xd2x
=-1/2*cos2x+C\uff08C\u4e3a\u4efb\u610f\u5e38\u6570\uff09
\u6c42\u51fd\u6570f(x)\u7684\u4e0d\u5b9a\u79ef\u5206\uff0c\u5c31\u662f\u8981\u6c42\u51faf(x)\u7684\u6240\u6709\u7684\u539f\u51fd\u6570\uff0c\u7531\u539f\u51fd\u6570\u7684\u6027\u8d28\u53ef\u77e5\uff0c\u53ea\u8981\u6c42\u51fa\u51fd\u6570f(x)\u7684\u4e00\u4e2a\u539f\u51fd\u6570\uff0c\u518d\u52a0\u4e0a\u4efb\u610f\u7684\u5e38\u6570C\u5c31\u5f97\u5230\u51fd\u6570f(x)\u7684\u4e0d\u5b9a\u79ef\u5206\u3002
\u6269\u5c55\u8d44\u6599\uff1a
\u5206\u90e8\u79ef\u5206\uff1a
(uv)'=u'v+uv'\uff0c\u5f97\uff1au'v=(uv)'-uv'\u3002
\u4e24\u8fb9\u79ef\u5206\u5f97\uff1a\u222b u'v dx=\u222b (uv)' dx - \u222b uv' dx\u3002
\u5373\uff1a\u222b u'v dx = uv - \u222b uv' d,\u8fd9\u5c31\u662f\u5206\u90e8\u79ef\u5206\u516c\u5f0f\u3002
\u4e5f\u53ef\u7b80\u5199\u4e3a\uff1a\u222b v du = uv - \u222b u dv\u3002
\u5e38\u7528\u79ef\u5206\u516c\u5f0f\uff1a
1\uff09\u222b0dx=c
2\uff09\u222bx^udx=(x^(u+1))/(u+1)+c
3\uff09\u222b1/xdx=ln|x|+c
4\uff09\u222ba^xdx=(a^x)/lna+c
5\uff09\u222be^xdx=e^x+c
6\uff09\u222bsinxdx=-cosx+c
7\uff09\u222bcosxdx=sinx+c
8\uff09\u222b1/(cosx)^2dx=tanx+c
9\uff09\u222b1/(sinx)^2dx=-cotx+c
10\uff09\u222b1/\u221a\uff081-x^2) dx=arcsinx+c
=(1/2)∫sin2xd(2x)
=-(1/2)cos2x + C
绛旓細杩囩▼濡備笅锛鈭玸in^2xdx sin^2x=(1-cos2x)/2 鈭玸in^2xdx =1/2鈭1dx-1/2鈭玞os2xdx =x/2-1/4鈭玞os2xd2x =x/2-sin2x/4+C
绛旓細绉垎杩囩▼涓 浠 = sin胃锛屽垯dx = cos胃 d胃 鈭垰(1-x²)dx =鈭垰(1-sin²胃)(cos胃 d胃)=鈭玞os²胃d胃 =鈭(1+cos2胃)/2d胃 =胃/2+(sin2胃)/4+C =(arcsinx)/2+(sin胃cos胃)/2 + C =(arcsinx)/2+(x鈭(1 - x²))/2+C =(1/2)[arcsinx...
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绛旓細涓ら閮芥病鏈夐敊锛屽彲浠ュ悎鑰屼负涓鐨勩俢os2x=1-2sin²x -1/2cos2x=sin²x-1/2 鍥犱负鏄涓嶅畾绉垎锛屾瘡涓粨鏋滃悗闈㈤兘鏄+甯告暟C鐨勶紝鍙渶瑕佺涓涓紡瀛愮殑C1-1/2鍜岀浜屼釜缁撹鐨凜2鐩哥瓑灏卞彲浠ヤ簡銆傚笇鏈涘浣犳湁鎵甯姪 濡傛湁闂锛屽彲浠ヨ拷闂傝阿璋㈤噰绾 绁濆涔犺繘姝 ...
绛旓細鈭玸in2xdx鐨鍘熷嚱鏁涓猴紙-1/2锛塩os2x+C銆侰涓虹Н鍒嗗父鏁般傝В绛杩囩▼濡備笅锛氭眰sin2x鐨勫師鍑芥暟灏辨槸瀵箂in2x杩涜涓嶅畾绉垎銆傗埆sin2xdx =锛1/2锛夆埆sin2xd2x =锛-1/2锛塩os2x+C 姝ｅ鸡鏄垹伪锛堥潪鐩磋锛夌殑瀵硅竟涓庢枩杈圭殑姣旓紝浣欏鸡鏄垹伪锛堥潪鐩磋锛夌殑閭昏竟涓庢枩杈圭殑姣斻傚嬀鑲″鸡鏀惧埌鍦嗛噷銆傚鸡鏄渾鍛ㄤ笂涓ょ偣杩炵嚎銆傛渶...
绛旓細鈭玸in2xdx=1/2鈭玸in2xd(2x)=-(1/2)cos2x+c(c涓轰换鎰忓父鏁)銆傛崲鍏冪Н鍒嗘硶鏄眰绉垎鐨勪竴绉嶆柟娉曪紝涓昏閫氳繃寮曡繘涓棿鍙橀噺浣滃彉閲忔浛鎹娇鍘熷紡绠鏄擄紝浠庤屾潵姹傝緝澶嶆潅鐨涓嶅畾绉垎銆傚畠鏄敱閾惧紡娉曞垯鍜屽井绉垎鍩烘湰瀹氱悊鎺ㄥ鑰屾潵鐨勩傚畾涔 鎹㈠厓绉垎娉曟槸姹傜Н鍒嗙殑涓绉嶆柟娉曪紝瀹冩槸鐢遍摼寮忔硶鍒欏拰寰Н鍒嗗熀鏈畾鐞嗘帹瀵艰屾潵鐨勩
绛旓細鈭1/SinxCosxdx=ln涓╰anx涓+C锛孋鏄Н鍒嗗父鏁般傝В绛杩囩▼濡備笅锛歝osxsinx=1/2脳sin2x锛岀悊鐢辨槸sin2x=2sinxcosx锛屼簩鍊嶈鍏紡銆涓嶅畾绉垎鐨勬ц川 涓嶅畾绉垎鏄竴涓嚱鏁伴泦鍚堬紝闆嗗悎涓嶅悓鐨勫厓绱犱箣闂寸浉宸竴涓浐瀹氱殑甯告暟銆傛牴鎹墰椤-鑾卞竷灏艰尐鍏紡锛岃澶氬嚱鏁扮殑瀹氱Н鍒嗙殑璁＄畻灏卞彲浠ョ畝渚垮湴閫氳繃姹備笉瀹氱Н鍒嗘潵杩涜锛岃繖閲岃娉ㄦ剰...
绛旓細=(1/2)鈭玿dsin2x =(1/2)xsin2x -(1/2)鈭玸in2xdx =(1/2)xsin2x +(1/4)cos2x + C 涓嶅畾绉垎鐨勬剰涔夛細璁綠(x)鏄痜(x)鐨勫彟涓涓鍘熷嚱鏁锛屽嵆∀x鈭圛锛孏'(x)=f(x)銆備簬鏄痆G(x)-F(x)]'=G'(x)-F'(x)=f(x)-f(x)=0銆傜敱浜庡湪涓涓尯闂翠笂瀵兼暟鎭掍负闆剁殑鍑芥暟蹇呬负甯告暟锛...
绛旓細鎵浠鈭玸in²xdx=锛1/2锛夆埆锛1-cos2x锛塪x=锛1/2锛夛紙x-锛1/2锛sin2x锛+C=0.5x-0.25sin2x+C銆涓嶅畾绉垎锛氬湪寰Н鍒嗕腑锛屼竴涓嚱鏁癴鐨勪笉瀹氱Н鍒嗭紝鎴鍘熷嚱鏁锛屾垨鍙嶅鏁帮紝鏄竴涓鏁扮瓑浜巉鐨勫嚱鏁癋锛屽嵆F鈥=f銆傝屼笉瀹氱Н鍒嗗拰瀹氱Н鍒嗛棿鐨勫叧绯荤敱寰Н鍒嗗熀鏈畾鐞嗙‘瀹氥傚叾涓璅鏄痜鐨勪笉瀹氱Н鍒嗐
绛旓細鈭玸in²xdx= 1/2x -1/4sin2x + C銆侰涓绉垎甯告暟銆傝В绛杩囩▼濡備笅锛氭牴鎹笁瑙掑叕寮 sin²x = (1-cos2x) / 2锛屽彲寰楋細鈭 sin²x dx = (1/2) 鈭 (1-cos2x) dx = (1/2) ( x- (1/2)sin2x) + C = 1/2x -1/4sin2x + C ...
